Chapter 10 - Conics, Parametric Equations, and Polar Coordinates - 10.1 Exercises - Page 692: 38

$${\text{Vertices}}\left( {2, - 12} \right){\text{ and }}\left( {2,12} \right){\text{ }}$$ $${\text{Asymptotes}}:{\text{ }}y = \pm 4\left( {x - 2} \right)$$ $${\text{Center}}\left( {2,0} \right)$$

$$\eqalign{ & {y^2} - 16{x^2} + 64x - 208 = 0 \cr & {\text{Group terms}} \cr & {y^2} - \left( {16{x^2} - 64x} \right) = 208 \cr & {y^2} - 16\left( {{x^2} - 4x} \right) = 208 \cr & {\text{Completing the square}} \cr & {y^2} - 16\left( {{x^2} - 4x + 4} \right) = 208 - 16\left( 4 \right) \cr & {y^2} - 16{\left( {x - 2} \right)^2} = 144 \cr & {\text{Divide the equation by 144}} \cr & \frac{{{y^2}}}{{144}} - \frac{{{{\left( {x - 2} \right)}^2}}}{9} = 1 \cr & \cr & {\text{This equation is written in standard form}} \cr & \frac{{{{\left( {y - k} \right)}^2}}}{{{a^2}}} - \frac{{{{\left( {x - h} \right)}^2}}}{{{b^2}}} = 1 \cr & {\text{Therefore,}} \cr & \frac{{{y^2}}}{{144}} - \frac{{{{\left( {x - 2} \right)}^2}}}{9} = 1 \to \frac{{{y^2}}}{{{{\left( {12} \right)}^2}}} - \frac{{{{\left( {x - 2} \right)}^2}}}{{{{\left( 3 \right)}^2}}} = 1 \cr & \underbrace {\frac{{{y^2}}}{{{{\left( {12} \right)}^2}}} - \frac{{{{\left( {x - 2} \right)}^2}}}{{{{\left( 3 \right)}^2}}} = 1}_{\frac{{{{\left( {y - k} \right)}^2}}}{{{a^2}}} - \frac{{{{\left( {x - h} \right)}^2}}}{{{b^2}}} = 1} \cr & h = 2,{\text{ }}k = 0,{\text{ }}a = 12,{\text{ }}b = 3 \cr & c = \sqrt {{a^2} + {b^2}} = \sqrt {144 + 9} \Rightarrow c = \sqrt {153} \cr & {\text{Characteristics of the hyperbola }} \cr & {\text{Orientation}}:{\text{ Vertical transverse axis}} \cr & {\text{Center}}\left( {h,k} \right) \to {\text{Center}}\left( {2,0} \right) \cr & \underbrace {{\text{Vertices}}\left( {h,k - a} \right){\text{ and }}\left( {h,k + a} \right)}_ \Downarrow \cr & {\text{Vertices}}\left( {2, - 12} \right){\text{ and }}\left( {2,12} \right){\text{ }} \cr & \underbrace {{\text{Foci}}\left( {h,k - c} \right){\text{ and }}\left( {h,k + c} \right)}_ \Downarrow \cr & {\text{Foci}}\left( {2, - \sqrt {153} } \right){\text{ and }}\left( {2, + \sqrt {153} } \right) \cr & \underbrace {{\text{Asymptotes}}:{\text{ }}y = \pm \frac{a}{b}\left( {x - h} \right) + k}_ \Downarrow \cr & {\text{Asymptotes}}:{\text{ }}y = \pm 4\left( {x - 2} \right) \cr & {\text{Graph:}} \cr} $$